Clifford and Euclidean translations of circles

We consider surfaces in the three-sphere that admit at least two circles through a generic closed point. Such surfaces are called "celestials" and can be obtained by moving a circle along a closed loop in at least two different ways. The radius of a circle is in general allowed to change during its motion through the three-sphere, however in this talk we consider the case where the radius remains fixed. Moreover, the motions we consider are translations. A translation is an isometry where every point moves with the same distance. We consider the projective three-sphere as a Cayley-Klein model for both elliptic geometry (aka Clifford geometry) and Euclidean geometry. For example the Clifford torus is well-known to be the Clifford translation of a great circle along a great circle. The goal of this presentation is to answer the following question:
Can a celestial in the three-sphere be—up to Moebius equivalence—both the Clifford- and Euclidean- translation of a circle?
For this purpose we propose elliptic and Euclidean invariants for surfaces in the spherical models. If time permits we give an overview of recent characterizations of celestials in terms of these invariants.